D. Schertzer, ENPC, Paris

S. Lovejoy, Physics, McGill

Part 2: Introduction to scaling in precipitation and hydrology

Dam Safety Interest Group,Expert’s meeting 9 November, 2004

Overview

• Fractal Sets, dimensions

• Co-dimensions

• Spectra

• Examples from hydrology

• Nonclassical Probability distributions

Some Classical Fractal sets and their dimension

– Cantor set• Cantor square

• Devil’s staircase

– Koch snowflake– Sierpinski Triangle

• Sierpinski Pyramid

– Peano curve

Cantor set

• Let us start with:

and let us iterate:

Koch snowflake

Let us start with:

and let us iterate:

Fractal Dimension• THE SIMPLEST METHOD

– take advantage of self-similarity. – a 1-dimensional line segment with the

magnification of 2, yields 2 identical line segments,

– a 2-dimensional (e.g.square or a triangle), with the magnification of 2, yields 4 identical shapes,

– a 3-dimensional cube, magnify it 2 times. you will get 8 identical cubes,

Fractal Dimension (2)

• Let’s use a variable D for dimension, for magnification, and N for the number of identical shapes:

» DN=

• Or:

• D = log N / log

First Examples

– Cantor Set:

D = log N / log log 2 / log ≈ 0.63

-Koch Snowflake:

D = log N / log log 4 / log ≈1.26.

– Daily Rain Fall Events in Dedougou (SW Africa):1922-1966

– Each line is a different year,

– eachblack point a rainy day.

– Cantor-like set– D≈log(7)/log(12):

• i.e. divide into 12 parts keep only 7..

– (Hubert et Carbonnel, 1990)

Rain Fall Events

Cantor Square

D= log 4/ log 3≈1.26

CASCADES

• Isotropic Cascade. The left hand side shows an non-intermittent (“homogeneous”) cascade, the right hand side shows how intermittency which can be modeled by assuming that sub-eddies are either “alive” or “dead” (model”).

L

LL/2

L/2

L/2

L/2

N(L) ã L- D N(L) ã L- Ds

Ln 4

Ln 2Ln 3™™

D = ™™Ln 2

= 2

D =s= 1.58

ISOTROPIC

= self similarity

-model

•The -model for   2, C = 0.2. •the set of the surviving “active” regions has a dimension equal to D= 2-C = 1.8.•the cascade process is iterated an infinite number of times, here it is followed for only four generations on a 256  256 point grid,

•Novikov and Steward (1963), Mandelbrot (1974), Frisch et al (1978)

Sierpinski Triangle

• or :

Sierpinski Triangle (2)

• D= log 3 /log 2 ≈ 1.58.

Sierpinski Pyramid

• First iteration:

D = log 4/ log 2 But DT = 1

10 th iteration:

Topological Dimension

– Definition:• the empty set Ø has topological dimension -1. • the topological dimension of a set is DT if and only if you can

disconnect it (by cutting it) by taking out a subset of topological dimension DT-1.

– Classical examples:» isolated point(s) DT = 0 can be cut only by Ø» lines DT =1 can be cut by an isolated point» Surfaces DT = 2 can be cut by a line.

– indeed a topological invariant, i.e. invariant under 1:1 and bicontinuous transformations.

Peano Curve

• motif:

D=log 9 / log 3 = 2i.e. it is a plane filling line

iterations:

A model of hydraulic network, From Steinhaus 1962

Geometric Method•The similarity method is

great for a fractal composed of a certain number of identical versions of itself…

•A way out: graph log(size) against log(magnification),

•details add additional irregularities, which add to the measurement.

• Fractal dimension = slope

Cloud perimeters over 5 decades yield D≈1.35 (Lovejoy, 1982)

Isotropic Scale Invariance and fractal sets

Fractal Dimension:

L

d=dimension of spaceD= fractal dimension of set

C=d-D= fractal codimension

Scale invariance:

Same form after zoom by factor .

n L( ) ∝ L

D

ρ L( ) =

n L( )

L

d

= ∝ L

D − d

= L

− C

n L( ) =

D

n L( ) D=scale invariant

Box Counting Dimension

• Sketch of the lava flow field from Etna (1900-1974) using box-counting technique:

• resolution is decreased by factors of 2 at each step. The finest resolution was 43 m.

• From Gaonac'h et al (1992).

N( λB ∩ A)~ −Dg(A)λ

“Defined” as the scaling exponent of the number of (nearly) disjoint boxes necessary to cover A:

Better understood as a crude “approximation” of the Hausdorff dimension

Are classical geostatistics Applicable to rain?

A

B C D

E F G

NT (L) ≈ −D(T)L

A) the field is shown with two isolines that have thresholds values; the box size is unity. In B), C) and D), we cover areas whose value exceeds by boxes that decrease in size by factors of two. In E), F) and G) the same degradation in resolution is

applied to the set exceeding the threshold.

-Classical geostatistics: D(T)=2-Monofractal: D(T)=const <2 , -Multifractal: D(T)<2,

decreasing function

Test using functional box

counting

Area =LimL→ 0 L2NT (L)( )

≈LimL→ 02−D(T )L → 0;

D T( ) < 2

Functional Box counting on 3D radar rain scans

102101100

L

101

102

103

104

N(L

)

101

L

101

102

103

104

105

N(L

)

reflectivity thresholds increasing (top to bottom) by factors of 2.5 (dat from Montreal).

NT (L) ≈ −D(T)L

Log10 N(L)

Log10 L

horizontal

Vertical and horizontal

Science: Lovejoy, Schertzer and Tsonis 1987

100km1km 1km 10km

Classical geostatistics

Functional box counting on French topography: 1 -1000km

103102101100100

101

102

103

104

105

106

L

N(L)

N(L) = number of covering boxes for exceedance sets at various altitudes. The dimensions d increase from 0.84 (3600m) to 1.92 (at 100m).

3600m

1800m100m

km

N(L)L-D

Multifractal: slopes vary with threshold

Lovejoy and Schertzer 1990

Slope =2 (required for classical geostatistics -regularity of Lebesgue measures)

Systematic resolution dependence

Fractal Codimension– Geometric definition:

• natural extension of vector subspace codimension

• If the set is A included in E (embedding space): D(A)=dim(A)< dim(E)=d

• Geometrical codimension: Cg (A)=d-D(A)

– As a consequence Cg is bounded:

• O≤Cg (A)≤d

Ex. In 3D space (dim(E)=3), the codimension of a line (dim(A)=1) is: Cg=3-1=2

Fractal Codimension (2)– stochastic processes:

• Probability of events, not the number of occurences

– Statistical definition:

• Codimension = scaling exponent of the

probability that a -dimensional ball of

resolution coversintersects A

Pr(Bλ ∩ A)=λ−c(A)

Pr(B3λ

∩ A)Pr(Bλ ∩ A)

=23

=3−C(A) ⇒ C(A)=log(3/2)log(3)

=1−log(2)log(3)

Example of the Cantor Set

Fractal Codimension (3)

Pr( λB ∩ A)~N( λB ∩ A)N( λB ∩ E)

~ −Dg(A)λ

−dλ

C(A)>d⇒ C(A) >Cg(A) (≡d)C(A)>d

D(A)≡d−C(A)⎫ ⎬ ⎭

⇒ D(A)<0

gC (A)<d=dim(E) <∞⇒ Cg(A)≡C(A)

• Relating the two definitions:

‘ latent dimension ’ paradox, in fact a statistical exponent !

Unbounded codimension

bounded codimension

Meteorological measuring network

L

Fractal set: each point is a station

9962 stations (WMO)

Number

n L( ) ∝ LD

Density ρ L( ) = n L( )L−2 ∝ L−C ; C = d − D; d = 2

The fractal dimension of the network=

1.75

Slope=D=1.75

L et al 86

C=2-1.75=0.25

Intersection Theorem

• if independent:

E1 ⊂E E2 ⊂E

C(E1 ∩ E2)=C(E1)+C(E2)

Pr(E1 ∩ E2) =Pr(E1)Pr(E2)

a trivial consequence of:

no trivial results for geometrical codimensions !

Ex. Sparse but violent regions of storms - no matter how “large” - with D<0.25 cannot be detected by the global network

Consequence: if C (E1 ∩E2 ) > D then the intersection is almost surely empty

Energy Spectra

Correlation functions, structure functions

For stationary processes X (statisticallytranslationally invariant along the time axis), wedefine

R τ( ) = X t( ) X t − τ( )

S τ( ) = X t( ) − X t − τ( )( )

2

R( )t is the correlation function, S( )t is thestructurefunctionandtherandomprocessX( )t isnonzerooverthe intervalfrom- /T 2to+ /T 2.The y arerelate d b:y

S τ( ) = 2 R 0( ) − R τ( )( )

wh en R(τ),R(0)diverge, (S τ)ca nconverge.

tT/2-T/2

X(t)

Spectral densities

%X ω( ) = X t( )−∞

∞

∫ e−iωtdt

P ω( ) =

1T

%X ω( )2

in 1-D "spectral density" (e.g. time) is defined as:

in D dimensions (e.g. space):

%X k( ) = X r( )∫ e−i k⋅rdr

P k( ) =

1N

%X k( )2

The(isotropic)

“spectrum”E k( ) = P k( )dD k

k=k∫

In D dimensions:

E ω( ) =P ω( ) + P −ω( ) =2P ω( )

In 1-D:

k = kwhere

Wiener-Khintchin Theorem

P k( ) = R r( )−∞

∞

∫ e−ik⋅rdr

This is the "Wiener Khintchin theorem" which relates the spectral density to autocorrelation function of a stationary process.

Tauberian theorem

E ω( ) ≈ω−

POWER LAWS F.T. POWER LAWS

S τ( ) ≈τ 2H ; H = −1( ) / 2

Note this is valid for 1<<3 (0≤H≤1) for S(τ), 1< (H<0) for R(τ).

Fourier scaling:

Structure function scaling:

Spectra in hydrology

1m

QuickTime™ and aAnimation decompressor

are needed to see this picture.

f295, 11293 drops

The angle averaged drop spectra5 storms, 18 triplets

Top: f142, 2nd= f145, 3rd=f295, 4th=f229, 5th=f207 thick line has theoretical slope: -2-5/3

1m-1

White noise(standard theory)

Corrsin-Obukov passive scalar theory

E k( ) ≈k−5/

Temporal Scaling of radar rain reflectivities

Temporal spectrum of the radar reflection from a single 30X27X37m pulse volume at 1km altitude. ω is in Hz.

(Duncan 1993)

E k( ) ≈k−5/

Hourly Rainfall

11 Years of hourly rainfall in Assink.De Lima 1998.

Synoptic maximum

Log10 frequency (hr-1)

Log

10

En e

r gy

0.00001 0.0001 0.001 0.01 0.1 1 r-

.

0.01

0.10.2

7

6

5

4

3

log

10

E(

ω

)

-.5 -.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

log10

ω (days

-1

)

16days

1Year

Temporal scaling and the Synoptic maximum

6.0

5.5

5.0

4.5

4.0

3.5

3.0

log

10

E(

ω

)

-.5 -.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

log10

ω (days

-1

)

16days

1Year

16 days = « synoptic maximum » = the lifetime of planetary size structures.

Tessier, Lovejoy, Schertzer, 1996

2 days11 years

RAIN

RIVER FLOWAverage daily river flow and rain for 30 French stations (catchments< 200km2).

a= Mississippib=Susquahanac=Arkansas d= Osagee=Colorado f=McCloudg=North Nashuah=Milli= Pendeltonj=Rocky Brook

Synoptic maximum

Pandey, Lovejoy, Schertzer, 1998

Spectra of

rivers from 1 day to

70 years

Ensemble average

Low frequency rain spectra

Fraedrich and Larnder 1993

=0.5

The average spectrum from 13 stations in Germany (daily precip)

Annual peak

Synoptic maximum

Climate: Northern Hemisphere average temperatures

From Lovejoy and Schertzer 1985

=1.8

Scaling of paleotemperatures: GRIP Greenland Ice Core

2001501005000

-44

-42

-40

-38

-36

-34

-32

-30

Figure 1

Time (kyr BP)

δΟ18

(%)

0,50,0-0,5-1,0-1,5-2,0-2,5

1

2

3

4

5

6

Figure 2

log (f) (kyr)

log (E(f))

f

-1.4

10

10

-1

High resolution (200 yr average) record the GRIP Greenland ice

core (Johnsen et al., 1992; GRIP members, 1993; Dansgaard et al.,

1993): •3,000 m long, 1,200 data points • sharp fluctuations at small time scales.

The power spectrum of the data (log-log plot); •global straight line is an indication of scaling. •no obvious frequency at (20 kyr)-1 or (40 kyr)-1

Schmitt, Schertzer Lovejoy 1995

Spatial Spectrum of radar reflectivity

k-1.45

Horizontal spectrum of 256X256 (McGill) radar scan with 75m resolution (from Tessier et al 1993).

Montreal Clouds: spectra

10

15

20

25

30

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

log E(k)

log k (1/m)

10

10

5

10

15

20

25

30

35

-3.5 -3 -2.5 -2 -1.5 -1 -0.5

log E(k)

log k (1/m)

10

10

Spectra of the smaller scenes, seperated in the vertical for clarity, with power law regressions shown

Sachs, Lovejoy, Schetzer 2002

Larger scenes

What is the outer scale of atmospheric turbulence?

Spectra of hundreds of satellite images spanning the scale range 1-5000 km, and 38 clouds spectra (1m-1km) from ground camera.

A multifractal analysis is more informative (see below).

Leff > 5 000 km !!

7

8

9

10

11

12

13

14

15

-4 -3 -2 -1 0 1 2 3

log E(k)10

log k (1/km)

AVHRR 12

GMS 5

SPOT

Photography

10

Lovejoy and Schertzer 2002

Topo-graphy

Spectrum

(20000km)-1 (1m)-1

Energy spectra over a scale range of 108 Global (ETOPO5, 10km), continental US (GTOPO30: 1km and 90m), Lower Saxony, 20cm).

Inadequate dynamic range

Slope -1.8

Fractional Integration and Differentiation

Brownian motion

Brownian motion X(t) is given by:dX

dt

= p ( t ) X t( ) = p ′t( ) d ′t

− ∞

t

∫

p(t) is a white noise.

In Fourier space:

i ω

˜

X ( ω ) =˜

p ( ω )

The structure function:

S τ( ) = τ σ

2

Fractional Integration, Fractional Brownian Motion

Consider the generalization:

d

s

X

s

dt

s

= p ( t )

withsoluti :on

X

s

t( ) = I

s

p t( )( )

whereIsisthefractionalintegral(s>0)derivative( <s 0) of the

function .pInfourierspace this isparticularlysimple:

i ω( )

s

˜

X

s

t( ) =˜p ω( )

˜

X

s

t( ) =˜p ω( ) i ω( )

− s

henc e for thespectr :um

E

Xs

ω( ) = E

p

ω( ) ω

− 2 s

= σ

2

ω

− 2 s

Real space properties

The structure function of Xs is obtained by the Tauberian theoremas above:

S τ( ) = τ

2 H

withH=-1s /2.For thesolutionXs(t):

X

s

t( ) = p t( ) ∗

Θ t( )

t

1 − s

⎡

⎣

⎢

⎤

⎦

⎥

whereΘ t( ) istheHeavisidefunction(=1,t<0,=0, ≥0t )andwehaveusedthetheconvol ution theorem ontheequation

˜

Xs

t( ) = ˜p ω( ) i ω( )

− s

and:

Thi s yields:

i ω( )

− s

→

F . T .

1

Γ s( )

Θ t( )

t

1 − s

⎡

⎣

⎢

⎤

⎦

⎥

X

s

t( ) =

1

Γ s( )

p ′t( )

t − ′t( )

s − 1

d ′t

− ∞

t

∫

Dimensions and spectraFor monofractal functions:

C=H=(-1)/2

So that for surfaces defined by d dimensional processes:

Dsurf=d+1-C=d+(3-)/2

Although this relation has been frequently used to estimate Dsurf

from , it is only valid for monofractals

Probability Distributions

Pr x > s( ) = p x( )dxs

∞

∫

Pr x > s( ) < e−as; s>>1

Pr x > s( ) ≈s−qD ; s>>1

(Tail) Cumulative Distribution Function

“Thin tailed” distributions

“Fat tailed” distributions (e.g. “Pareto” / power law)

Moments xq = xqdPr

−∞

∞

∫ xq → ∞; q≥qD

Hydrometeorological long time series

Return period 100 years (algebraic law)

Return period 1000 years (exponential law)

Padova series (Italy): empirical probability distribution (dots), normal fit (continouous line) and aymptotic power-law (dashed line).

Bendjoudi et Hubert Rev. Sci. Eau,1999

qD ≈4

Radar reflectivity of rain:

Probability distributions

Probability distribution of radar reflectivities from 10 constant altitude maps (resolution varying from 0.25 to 2.5 km, range 20 to 200km).

qD=1.06

From Schertzer and Lovejoy 1987.

French rivers and precipitation (<200km2)

-6

-5

-4

-3

-2

-1

0

210-1

log Pr (P' > P)

10

log P

10

210-1

-6

-5

-4

-3

-2

-1

0

log Q

10

log Pr (Q' > Q)

10

Prob of a daily river flow Q' exceeding Q from 30 time series from the corresponding river.

Prob of a daily rainfall accumulation P' exceeding P from 30 time series,

France.qD = 2.7

qD = 3.6

Pr ′P > P( )≈P−qD

PqD → ∞; q>qD

French river, small basins Tessier, L+S 1996

Probability distributions of Normalized US Rivers: Divergence of moments

Pr ′Q >Q( ) ≈Q−qD

QqD → ∞; q> qDExtreme events

20 US rivers with basins in the range 4-106 km2; 10-75 years in length

Pandey, Lovejoy, Schertzer 1998

Temperature distributions, northern hemisphere

4 years

16 years

64 years

qD=5

Lovejoy and Schertzer 1985 (data from Jones et al 1982)

Temperature probability distributions for paleotemperatures

qD=5

350 years

1400 years

5600 years22400 years

Lovejoy and Schertzer 1985 (data from Greenland Camp Century core)

Conclusions1. Scaling/scale invariant sets are fractalsFractal dimensions and codimensions are scaling exponents

2. Scaling fields multifractals, spectral analysis.

3. Rain, temperature, topography, river flow show wide range scaling.

4. Probabilities can have “fat tails”: slow, algebraic fall-off.